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求一算術

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Text ID
KR3fc041
Date
1800–1810
Dynasty
清

Qiúyī suànshù 求一算術

Mathematical Method of the Seeking-One Procedure by 張敦仁 (撰)

About the work

張敦仁 Zhāng Dūnrén’s (1754–1834) systematic QiánJiā exposition of 秦九韶 Qín Jiǔsháo’s Dàyǎn qiúyī shù 大衍求一術 (the Chinese Remainder Theorem), in 4 juàn. The first thorough Qīng exposition of this fundamental SòngYuán algorithm, preceding 焦循 Jiāo Xún’s KR3fc044 Tiānyuányī shì and forming part of the broader Yáng-zhōu-circle project of recovering the SòngYuán mathematical methods.

Abstract

The Dàyǎn qiúyī shù — Qín Jiǔsháo’s 1247 systematic algorithm for solving systems of linear congruences (now known as the Chinese Remainder Theorem) — is one of the great pre-modern Chinese mathematical achievements, but its presentation in the KR3fc008 Shùshū jiǔzhāng / KR3f0041 Shùxué jiǔzhāng is highly compressed. Qín Jiǔsháo had given the general algorithm together with worked examples, but had not provided a full conceptual exposition of why the algorithm works; by the late Míng the procedural details had become inaccessible to working mathematicians.

Zhāng Dūnrén’s task in the Qiúyī suànshù was the recovery of the substantive algorithm. The four juàn cover:

(1) The basic problem of linear congruences and the special case of pair-wise coprime moduli (where the algorithm is straightforward).

(2) The general case where the moduli share common factors — the harder case that Qín Jiǔsháo had systematically addressed but whose treatment had become obscure.

(3) The qiúyī (seeking-one) sub-procedure — the procedure for finding, for a given pair of coprime integers (a, b), the smallest positive integer x such that ax ≡ 1 (mod b). This is the conceptual heart of the algorithm; Zhāng Dūnrén supplies a full step-by-step exposition with worked examples.

(4) Applied problems: Qín Jiǔsháo’s original 81 problems retraversed with the recovered algorithmic clarity, plus additional Zhāng-supplied problems illustrating various edge cases and applications.

The work was the first systematic Qīng exposition of the qiúyī method and made the algorithm newly accessible to mid-Qīng working mathematicians. It is one of the principal documents of the Jiā-Dào-era SòngYuán mathematical recovery — the project, coordinated across the Yángzhōu mathematical circle, of restoring the lost algebraic methods of the SòngYuán tradition to active circulation. Zhāng Dūnrén’s qiúyī exposition is the foundational text on which subsequent QiánJiā treatments of the algorithm depend.

Dating: NotBefore set at 1800; notAfter at 1810, allowing for the work’s completion in the first decade of the nineteenth century.

Translations and research

  • Libbrecht, Ulrich. 1973. Chinese Mathematics in the Thirteenth Century: The Shu-shu chiu-chang of Ch’in Chiu-shao. Cambridge, Mass.: MIT Press. — Treats Zhāng Dūnrén’s recovery exposition in detail.
  • Lam Lay-Yong. 1986. “Linkages: Exploring the Similarities between the Chinese Rod Numeral System and Our Numeral System.” Archive for History of Exact Sciences 37: 365–392.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7.
  • Bréard, Andrea. 1999. Re-Kreation eines mathematischen Konzepts im chinesischen Diskurs. Stuttgart: Steiner. — Treats the Jiā-Dào-era qiú-yī recovery.

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